Real Submanifolds and Holomorphic Mappings in Geometric Function Theory

几何函数理论中的实子流形和全纯映射

• 批准号: 0305474
• 负责人: Xianghong Gong
• 金额: $ 11.07万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0305474&HistoricalAwards=false
• 关键词: Real; Submanifolds; Holomorphic; Mappings; Geometric

Abstract: The long-term goal of this project is to study holomorphicmappings and real submanifolds arising in complex analysis andholomorphic dynamical systems. A large part of the PI's proposedresearch has its roots in Birkhoff's fixed point theorem, the KAM(Kolmogorov-Arnold-Moser) theory, and the Siegel-Bruno-Yoccoztheory. On the other hand, new research obtained by the method ofcomplex analysis will give new insights into the classicalarea-preserving maps and reversible maps, and reversible orHamiltonian systems. The PI's proposed research involves severalproblems in connection with real analytic surfaces intwo-dimensional complex space, formally linearizable reversible orarea-preserving real analytic maps, and singular Levi-flathypersurfaces in the complex projective space. The differential equations concerning the motion of N mass points, a model of the solar system,in the three-dimensional space attracting each other according to the Newton's law form a Hamiltonian system.The reversibility of a dynamical system could just be the time-symmetry that is important in practical matters. The periodic orbits of certain area-preserving mappings correspond to the periodic motion in the Hamiltonian system of the restricted three-body problem, and such studyof the existence of such periodic orbits goes back at least to work of Poincar\'e and Birkhoff on the celestial mechanics about a century ago. The PI's work aims to understand the existence of periodic orbits in such Hamiltonian or time-symmetric systems, both over the real numbers and over the complex numbers.The PI is active in undergraduate teaching, andhas also organized a graduate student seminar dealing with his research area.

摘要：该项目的长期目标是研究复杂分析和全型动力学系统中产生的全态性和真实子膜。 PI提出的研究的很大一部分源于Birkhoff的固定点定理，KAM（Kolmogorov-Arnold-Moser）理论和Siegel-Bruno-Yoccoztheory。 另一方面，通过Complex Analysis获得的新研究将为经典的保护图和可逆地图以及可逆的Orhamiltonian Systems提供新的见解。 PI提出的研究涉及与实际分析表面上的几个问题，在复杂的空间中具有正式可可逆的Orarea提供的真实分析图以及复杂的投影空间中的奇异LEVI-FLATHYPERFACE。关于n个质量点的运动的微分方程，即太阳系的模型，在三维空间中，根据牛顿定律互相吸引。动态系统的可逆性可能是在实际问题中很重要的时间对称性。 某些保护区域的映射的周期性轨道对应于受限制的三体问题的哈密顿系统中的周期性运动，并且这种周期性轨道的存在至少可以追溯到大约一个世纪前在天体机械上的Poincar \ e和Birkhoff的工作。 PI的工作旨在了解这种哈密顿量或时间对称系统中的周期性轨道的存在，无论是在实数还是复杂数字上。